Communication between two parties across a communication channel, between a transmitter and receiver, are subject to unknowns such as noise and filtering by the channel's impulse response (CIR). The CIR of a communication channel characterizes the channel. The CIR is uniquely defined by a set of parameters (the set of parameters usually comprise amplitudes and times of flight of reflections of a transmitted impulse signal). With respect to decoding, the noise is treated as nuisance parameters, and the parameters of the CIR as unknowns which are to be estimated as precisely as possible to maximize the accuracy of decoding of transmitted signals.
The estimation of the parameters of the CIR is referred to in the art as “channel estimation”. In order to estimate the coefficients of the CIR, the communication channel can be used to transmit a signal known at both ends i.e. a pilot signal (pilots), to gain knowledge about the CIR. This dictates a trade-off between the portion of the communication channel reserved for the transmission of pilots (and thus lost to data transmission) and the decoding error rate due to bad channel estimation, both affecting the communication bitrate. Channel estimation therefore requires the selection of pilots and the design of an estimation algorithm used for estimating the channel parameters.
The pilots provide information about the CIR, and so does an a priori knowledge about the communication channel's model (such as whether the channel is multipath, flat-fading, scattering, and/or band-limited). In the ideal case wherein the communication channel is noiseless, when an impulse signal is passed from a transmitter to a receiver along the channel, the CIR can be perfectly recovered with a finite set of samples of the signal received at the receiver provided the channel perfectly obeys the a priori known channel model; thus a sampling theorem is defined. If the pilots give uniform samples in time at the Nyquist rate and if the CIR is band-limited with a bandwidth lower than, or equal to, the Nyquist rate, then the CIR can be perfectly reconstructed from the finite set of samples.
A multipath channel is a channel which has a CIR which comprises K components wherein K is an integer number greater than or equal to 1. Each of the K components corresponds to a reflection of the transmitted impulse signal from a scatterer. Each of the K components is defined by two parameters which are: a time of arrival of the reflection at a receiver and an amplitude of the reflection. Therefore, each CIR of a multipath channel is defined by K*2 parameters. The amplitudes may be a complex value and the time of flights may be real values.